Hindawi Publishing Corporation Advances in Mathematical Physics Volume 2011, Article ID 259089, 45 pages doi:10.1155/2011/259089
Research Article Dimensional Enhancement via Supersymmetry
M. G. Faux,1 K. M. Iga,2 and G. D. Landweber3
1 Department of Physics, State University of New York, Oneonta, NY 13820, USA 2 Natural Science Division, Pepperdine University, Malibu, CA 90263, USA 3 Department of Mathematics, Bard College, Annandale-on-Hudson, NY 12504-5000, USA
Correspondence should be addressed to M. G. Faux, [email protected]
Received 3 March 2011; Revised 24 May 2011; Accepted 24 May 2011
Academic Editor: Yao-Zhong Zhang
Copyright q 2011 M. G. Faux et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
We explain how the representation theory associated with supersymmetry in diverse dimensions is encoded within the representation theory of supersymmetry in one time-like dimension. This is enabled by algebraic criteria, derived, exhibited, and utilized in this paper, which indicate which subset of one-dimensional supersymmetric models describes “shadows” of higher-dimensional models. This formalism delineates that minority of one-dimensional supersymmetric models which can “enhance” to accommodate extra dimensions. As a consistency test, we use our formalism to reproduce well-known conclusions about supersymmetric field theories using one- dimensional reasoning exclusively. And we introduce the notion of “phantoms” which usefully accommodate higher-dimensional gauge invariance in the context of shadow multiplets in supersymmetric quantum mechanics.
1. Introduction
Supersymmetry 1–5 imposes increasingly rigid constraints on the construction of quantum field theories 6–11 as the number of space-time dimensions increases. Thus, there are fewer supersymmetric models in six dimensions than there are in four, and yet fewer in ten dimensions 12 . In eleven dimensions there seems to be a unique possibility 13 ,at least on-shell. Anomaly freedom imposes seemingly distinct algebraic constraints which make this situation even more interesting. However, the off-shell representation theory for supersymmetry is well understood only for relatively few supersymmetries, and remains a mysterious subject in contexts of special interest, such as N 4 Super Yang Mills theory, and the four ten-dimensional supergravity theories 14 . Many lower-dimensional models can be obtained from higher-dimensional models by dimensional reduction 15, 16 . Thus, a subset of lower-dimensional supersymmetric 2 Advances in Mathematical Physics theories derives from the landscape of possible ways that extra dimensions can be removed. But most lower-dimensional theories do not seem to be obtainable from higher-dimensional theories by such a process; they seem to exist only in lower-dimensions. We refer to a lower- dimensional model obtained by dimensional reduction of a higher-dimensional model as the “shadow” of the higher-dimensional model. So we could rephrase our comment above by saying that not all lower dimensional supersymmetric theories may be interpreted as shadows. It is a straightforward process to construct a shadow theory from a given higher- dimensional theory. But it is a more subtle proposition to construct a higher-dimensional supersymmetric model from a lower-dimensional model, or to determine whether a lower dimensional model actually does describe a shadow, especially of a higher-dimensional theory which is also supersymmetric. We have found resident within lower-dimensional supersymmetry an algebraic key which provides access to this information. A primary purpose of this paper is to explain this. It is especially interesting to consider reduction to one time-like dimension, by switching off the dependence of all fields on all of the spatial coordinates. Such a process reduces quantum field theory to quantum mechanics. Upon making such a reduction, in- formation regarding the spin representation content of the component fields is replaced with R-charge assignments. But it is not obvious whether the full higher-dimensional field content, or the fact that the one-dimensional model can be obtained in this way, is accessible information given the one-dimensional theory alone. As it turns out, this infor- mation lies encoded within the extended one-dimensional supersymmetry transformation rules. We refer to the process of restructuring a one-dimensional theory so that fields depend also on extra dimensions in a way consistent with covariant spin 1,D− 1 assignments and other structures, such as higher-dimensional supersymmetries, as “dimensional enhance- ment”. This process describes the reverse of dimensional reduction. We like to envision this in terms of the relationship between a higher-dimensional “ambient” theory, and the restriction to a zero-brane embedded in the larger space. A supersymmetric quantum mechanics then describes the “worldline” physics on the zero-brane. And the question as to whether this worldline physics “enhances” to an ambient space-time field theory is the reverse of viewing the worldline physics as the restriction of a target-space theory to the zero-brane. If the particular supersymmetric quantum mechanics obtained by restriction of a given theory to a zero-brane depended on the particular spin 1,D − 1 -frame described by that zero-brane, the higher-dimensional theory would not respect spin 1,D − 1 - invariance. Thus, if a one-dimensional theory enhances into a spin 1,D− 1 -invariant higher- dimensional theory, then the higher-dimensional theory obtained in this way should be agnostic regarding the presence or absence of an actual zero-brane on which such a one- dimensional theory might live. This observation, in conjunction with the requirement of higher-dimensional supersymmetry, provides the requisite constraint needed to resolve the enhancement question. In particular, by imposing spin 1,D− 1 -invariance on the enhanced supercharge operator, we are able to complete the ambient field-theoretic supercharge operator given merely the “time-like” restrictions of this operator. We find this interesting and surprising. The proposition that one can systematically delineate those one-dimensional theories which can enhance to higher-dimensions, and also discern how the higher-dimensional spin structures may be switched back on, is empowered by the fact that the representation theory of one-dimensional supersymmetry is relatively tame when compared with the Advances in Mathematical Physics 3 representation theory of higher-dimensional supersymmetry, for a variety of reasons. This enables the prospect of disconnecting the problem of spin assignments from the problem of classifying and enumerating supersymmetry representations, allowing these concerns to be addressed separately, and then merged together afterwards. With this motivation in mind, we have been developing a mathematical context for the representation theory of one- dimensional supersymmetry, also with other collaborators. The primary purpose of this paper is to demonstrate that the landscape of supersym- metry representation theory in any number of space-time dimensions resides fully-encoded within the seemingly-restricted regime of one-dimensional worldline supersymmetry representation theory. We find this result remarkable, compelling, and noteworthy, regardless of how complicated it may prove to algorithmically “extract” this information. But we demonstrate below that algorithms to perform such extractions do exist. In fact, we present explicit examples of algorithms which delineate one-dimensional models which are shadows of higher-dimensional models from those which are not. We do not purport that our algorithms are optimized. And we view this paper as a plateau from which more efficient algorithms could be developed. A cursory accounting of the complexity of the general problem is addressed in Section 5. In a sequence of papers 17–23 , we have explored the connection between repre- sentations of supersymmetry and aspects of graph theory. We have shown that elements of a wide and physically relevant class of one-dimensional supermultiplets with vanishing central charge are equivalent to specific bipartite graphs which we call Adinkras; all of the salient algebraic features of the multiplets translate into restrictive and defining features of these objects. A systematic enumeration of those graphs meeting the requisite criteria would thereby supply means for a corresponding enumeration of representations of supersymmetry. In 24, 25 , we have developed the paradigm further, explaining how, in the case of N-extended supersymmetry, the topology of all connected Adinkras are specified by quotients of N-dimensional cubes, and how the quotient groups are equivalent to doubly- even linear binary block codes. Thus, the classification of connected Adinkras is related to the classification of such codes. In this way we have discovered an interesting connection between supersymmetry representation theory and coding theory 26–28 . All of this is part of an active endeavor aimed at delineating a mathematically-rigorous representation theory in one-dimension. In this paper we use the language of Adinkras, in a way which does not presuppose a deep familiarity with this topic. We have included Appendix B as a brief and superficial primer, which should enable the reader to appreciate the entirety of this paper self- consistently. Further information can be had by consulting our earlier papers on the subject. In this paper we focus on the special case of enhancement of one-dimensional N 4 supersymmetric theories into four-dimensional N 1 theories. This is done to keep our discussion concise and concrete. Another motivating reason is because the supersymmetry representation theory for 4D N 1 theories is well known. Thus, part and parcel of our discussion amounts to a consistency check on the very formalism we are developing. From this point of view, this paper provides a first step in what we hope is a continuing process by which yet-unknown aspects of off-shell supersymmetry can be discerned. In the context of 4D theories, we use standard physics nomenclature, and refer to spin 1, 3 -invariance as “Lorentz” invariance. We should mention that the prospect that aspects of higher-dimensional supersymme- try might be encoded in one-dimensional theories was suggested years ago in unpublished work 29 by Gates et al. Accordingly, we had used that attractive proposition as a 4 Advances in Mathematical Physics prime motivator for developing the Adinkra technology in our earlier work. This paper represents a tangible realization of that conjecture. Complementary approaches towards resolving a supersymmetry representation theory have been developed in 30–36 . Other ideas concerning the relevance of one-dimensional models to higher-dimensional physics were explored in 37, 38 . This paper is structured as follows. In Section 2 we describe an algebraic context for discussing supersymmetry tailored to the process of dimensional reduction to zero-branes and, vice-versa, to enhancing one-dimensional theories. We explain how higher-dimensional spin structures can be accommodated into vector spaces spanned by the boson and fermion fields, and how the supercharges can be written as first-order linear differential operators which are also matrices which act on these vector spaces. This is done by codifying the supercharge in terms of diophantine “linkage matrices”, which describe the central algebraic entities for analyzing the enhancement question. In Section 3 we explain how Lorentz invariance allows one to determine “space- like” linking matrices from the “time-like” linking matrices associated with one-dimensional supermultiplets, and thereby construct a postulate enhancement. We then use this to derive nongauge enhancement conditions, which provide an important sieve which identifies those one-dimensional multiplets which cannot enhance to four-dimensional nongauge matter multiplets. In Section 4 we apply our formalism in a methodical and pedestrian manner to the context of minimal one-dimensional N 4 supermultiplets, and show explicitly how the known structure of 4D N 1 nongauge matter may be systematically determined using one-dimensional reasoning coupled only with a choice of 4D spin structure. We explain also how our nongauge enhancement condition provides the algebraic context which properly delineates the chiral multiplet shadow from its 1D “twisted” analog, explaining why the latter cannot enhance. In Section 6 we generalize our discussion to include 2-form field-strengths subject to Bianchi identities. This allows access to the question of enhancement to vector multiplets. In the process we introduce the notion of one-dimensional “phantom” fields which prove useful in understanding how gauge invariance manifests on shadow theories. We use the context of the 4D N 1 Abelian vector multiplet as an archetype for future generalizations. We also include five appendices which are an important part of this paper. Appendix A is especially important, as this provides the mathematical proof that imposing Lorentz invariance of the postulate linkage matrices allows one to correlate the entire higher- dimensional supercharge with its time-like restriction. We also derive in this appendix algebraic identities related to the spin structure of enhanced component fields, which should provide for interesting study in the future generalizations of this work. Appendix B is a brief summary of our Adinkra conventions, explaining technicalities, such as sign conventions, appearing in the bulk of the paper. Appendix C explains the dimensional reduction of the 4D N 1 chiral multiplet, complementary to the nongauge enhancement program described in Section 4. Appendix D explains the dimensional reduction of the 4D N 1 Maxwell field-strength multiplet, complementary to Section 5.This shows in detail how phantom sectors correlate with gauge aspects of the higher-dimensional theory. Appendix E is a discussion of four-dimensional spinors useful for understanding details of our calculations. We use below some specialized terminology. Accordingly, we finish this introduction section by providing the following three-term glossary, for reference purposes. Advances in Mathematical Physics 5
Shadow
We refer to the one-dimensional multiplet which results from dimensional reduction of a higher-dimensional multiplet as the “shadow” of that higher-dimensional construction.
Adinkra
The term Adinkra refers to 1D supermultiplets represented graphically, as explained in Appendix B. We sometimes use the terms Adinkra, supermultiplet, and multiplet synony- mously.
Valise
A Valise supermultiplet, or a Valise Adinkra, is one in which the component fields span exactly two distinct engineering dimensions. These multiplets form representative elements of larger “families” of supermultiplets derived from these using vertex-raising operations, as explained below. Thus, larger families of multiplets may be unpacked, as from a suitcase or a valise , starting from one of these multiplets.
2. Ambient versus Shadow Supersymmetry
It is easy to derive a one-dimensional theory by dimensionally reducing any given higher- dimensional supersymmetric theory. Practically, this is done by switching off the dependence of all fields on the spatial coordinates, by setting ∂a → 0. One way to envision this process is as a compactification, whereby the spatial dimensions are rendered compact and then shrunk to zero size. Alternatively, we may envision this process as describing a restriction of a theory onto a zero-brane, which is a time-like one-dimensional submanifold embedded in a larger, ambient, space-time. Using this latter metaphor, we refer to the restricted theory as the “shadow” of the ambient theory, motivated by the fact that physical shadows are constrained to move upon a wall or a wire upon which the shadow is cast.
2.1. Ambient Supersymmetry
Supersymmetry transformation rules can be written in terms of off-shell degrees of freedom, by expressing all fields and parameters in terms of individual tensor or spinor components. Thus, without loss of generality, we can write the set of boson components as φi and the set of fermion components as ψ ı, without being explicitly committal as the the spin 1,D − 1 - representation implied by these index structures. Generically, a spin 1,D− 1 -transformation acts on these components as